Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0

Let $K$ be a one-variable function field over a field of constants of characteristic 0. Let $R$ be a holomorphy subring of $K$, not equal to $K$. We prove the following undecidability results for $R$: If $K$ is recursive, then Hilbert's Tenth Problem is undecidable in $R$. In general, there exist $x_1,...,x_n \in R$ such that there is no algorithm to tell whether a polynomial equation with coefficients in \Q(x_1,...,x_n) has solutions in $R$.Comment: This version contains minor revisions and will appear in Annales de l Institut Fourie

Degree spectra for transcendence in fields

We show that for both the unary relation of transcendence and the finitary relation of algebraic independence on a field, the degree spectra of these relations may consist of any single computably enumerable Turing degree, or of those c.e. degrees above an arbitrary fixed $\Delta^0_2$ degree. In other cases, these spectra may be characterized by the ability to enumerate an arbitrary $\Sigma^0_2$ set. This is the first proof that a computable field can fail to have a computable copy with a computable transcendence basis

The King's many bodies: the self-deconstruction of law's hierarchy

The article connects two strands of the recent sociolegal debate: (1) the empirical discovery of new forms of spontaneous law in die Course of globalization, and (2) the emergence of deconstructive theories of law that undermine the law's hierarchy. The article puts forward the thesis that law's hierarchy has successfully resisted all old and new attempts at its deconstruction; it breaks, however, under the pressures of globalization that produced a global law without the state, as self-created law of global society that has no institutionalized support whatsoever in international poliucs and public international law. Consequently, the article criticizes deconstructive theories for their lack of autological analysis. These theories do not take into account the historical condicions of deconstruction. Accordingly, deconstructive analysis of law would have to look for new legal distinctions that are plausible under the new condicions of a doubly fragmented global society. The article sketches the contours of an emerging polycontextural law